Let $a\in\mathbb{R}^k$ be a random vector sampled from $N(0,\Sigma)$. Let $X = aa^T - \Sigma$. Then we have $\mathbb{E} X = 0$. Can we find a constant $C\in\mathbb{R}$ and another fixed matrix $A\in\mathbb{R}^{k\times k}$ such that $$ \mathbb{E} X^p \preceq \frac{p!}{2}C^{p-2}A^2? $$ The above property is the sub-exponential property, so I think it holds for matrix $X$.
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$\begingroup$ What does $\preceq$ stand for? $\endgroup$– Ben DeitmarCommented Jun 26 at 21:16
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$\begingroup$ Thanks for asking about the clarification. If matrices $A \preceq B$, then $A-B \preceq 0$ ($A-B$ is a negative-definite matrix). $\endgroup$– NicoleCommented Jun 27 at 4:22
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